Properties

Label 2-1053-9.7-c1-0-32
Degree $2$
Conductor $1053$
Sign $0.766 + 0.642i$
Analytic cond. $8.40824$
Root an. cond. $2.89969$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.500 + 0.866i)4-s + (−1 − 1.73i)5-s + (2 − 3.46i)7-s − 3·8-s + 1.99·10-s + (−2 + 3.46i)11-s + (−0.5 − 0.866i)13-s + (1.99 + 3.46i)14-s + (0.500 − 0.866i)16-s + 2·17-s + (0.999 − 1.73i)20-s + (−1.99 − 3.46i)22-s + (0.500 − 0.866i)25-s + 0.999·26-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.250 + 0.433i)4-s + (−0.447 − 0.774i)5-s + (0.755 − 1.30i)7-s − 1.06·8-s + 0.632·10-s + (−0.603 + 1.04i)11-s + (−0.138 − 0.240i)13-s + (0.534 + 0.925i)14-s + (0.125 − 0.216i)16-s + 0.485·17-s + (0.223 − 0.387i)20-s + (−0.426 − 0.738i)22-s + (0.100 − 0.173i)25-s + 0.196·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1053\)    =    \(3^{4} \cdot 13\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(8.40824\)
Root analytic conductor: \(2.89969\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1053} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1053,\ (\ :1/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.102250467\)
\(L(\frac12)\) \(\approx\) \(1.102250467\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.5 - 0.866i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2 + 3.46i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5 + 8.66i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6 + 10.3i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4 - 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2 - 3.46i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 + (5 - 8.66i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.799815301807426672559061395746, −8.663604449681589915151562912100, −7.961657803175276911099579320558, −7.52120848754557787243523229287, −6.81373758955691357679580602282, −5.50318358385417967926218792156, −4.52850970604890807957077009337, −3.80735421179063566397293340183, −2.26798651304069144346399973742, −0.57784854588684380191294648504, 1.40526707573869599306845730655, 2.69980677707273842439706379571, 3.21106786701521275019249777931, 4.97656881435697260778614980774, 5.70556292082506757176595982950, 6.55300868852639826595281104303, 7.63760159610305741771359912362, 8.599251330176785936745951053385, 9.096259295277297379348565894476, 10.25697650412863615088700128720

Graph of the $Z$-function along the critical line